Analytical solution of the one‐dimensional nonlinear Richards equation based on special hydraulic functions and the variational principle

Abstract

SummaryA simple analytical solution of a one‐dimensional transient and nonlinear Richards equation is of great importance for estimating the hydraulic properties of soil and for precision irrigation. We developed simple approximate solutions in explicit and implicit forms for equations describing one‐dimensional, constant‐water‐head horizontal absorption and vertical infiltration for a special case of an exponential water‐retention curve and power‐law hydraulic conductivity, respectively, based on least‐action and variational principles. The profile of the soil water content (SWC) depended on the soil hydraulic parameter n associated with the pore‐size distribution, and the new solutions could be applied to most soils for 3.5 ≤ n ≤ 24.5. A comparison with numerical solutions calculated by an implicit‐difference scheme indicated that the approximate solution in explicit form accurately estimated the special hydraulic‐function parameters for calculating the SWC profile, cumulative infiltration, infiltration rate and wetting‐front distance in horizontal‐absorption experiments. An approximate solution in implicit form was also obtained with Nth order Taylor‐series expansions for an equation describing one‐dimensional, constant‐water‐head vertical infiltration. A large N was used to simulate SWC for soils with a large n, but this did not influence estimates of cumulative infiltration. The relation between infiltration rate and the inverse wetting‐front depth was not strictly linear for vertical infiltration, and the implicit method could be used to describe this nonlinear relation.Highlights A simple solution of the Richards equation is important to research soil water infiltration. Least‐action and variational principles were used to analyse soil water content distribution. Analytical solutions of a nonlinear Richards equation by special hydraulic functions were proposed. A method was proposed to predict the infiltration and estimate soil hydraulic parameters.